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Plurals and Mereology
Salvatore Florio, David Nicolas
To cite this version:
Salvatore Florio, David Nicolas. Plurals and Mereology. Journal of Philosophical Logic, Springer Verlag, 2021, 50 (3), pp.415-445. �10.1007/s10992-020-09570-9�. �ijn_02978254v1�
https://doi.org/10.1007/s10992-020-09570-9
Plurals and Mereology
Salvatore Florio1 ·David Nicolas2
Received: 2 August 2019 / Accepted: 5 August 2020 /
©The Author(s) 2020
Abstract
In linguistics, the dominant approach to the semantics of plurals appeals to mere- ology. However, this approach has received strong criticisms from philosophical logicians who subscribe to an alternative framework based on plural logic. In the first part of the article, we offer a precise characterization of the mereological approach and the semantic background in which the debate can be meaningfully reconstructed.
In the second part, we deal with the criticisms and assess their logical, linguistic, and philosophical significance. We identify four main objections and show how each can be addressed. Finally, we compare the strengths and shortcomings of the mereologi- cal approach and plural logic. Our conclusion is that the former remains a viable and well-motivated framework for the analysis of plurals.
Keywords Mass nouns·Mereology·Model theory·Natural language semantics· Ontological commitment·Plural logic·Plurals·Russell’s paradox·Truth theory
1 Introduction
A prominent tradition in linguistic semantics analyzes plurals by appealing to mere- ology (e.g. Link [40,41], Landman [32,34], Gillon [20], Moltmann [50], Krifka [30], Bale and Barner [2], Chierchia [12], Sutton and Filip [76], and Champollion [9]).1
1The historical roots of this tradition include Leonard and Goodman [38], Goodman and Quine [22], Massey [46], and Sharvy [74].
Salvatore Florio [email protected] David Nicolas
1 Department of Philosophy, University of Birmingham, Birmingham, United Kingdom
2 Institut Jean Nicod, D´epartement d’´etudes cognitives, ENS, EHESS, CNRS, PSL University, Paris, France
The mereological approach to plural semantics has received strong criticisms from philosophical logicians who subscribe to an alternative framework based on plu- ral logic (e.g. Boolos [5], Oliver and Smiley [57,58], Rayo [65,66], Yi [85], and McKay [47]). Some of the criticisms target a broader class of “singularist” semantic analyses that interpret plural expressions in terms of singular ones. The mereological approach is the most popular, and perhaps the most plausible, of these analyses.
These criticisms have been very influential in philosophy, providing grounds for the acceptance of plural logic in areas such as metaphysics and the philosophy of mathematics. What has been overlooked is that, once the mereological approach is properly understood, its proponents have the basic tools for responding. Our aim is to clarify what these responses can be and develop a systematical defense of this approach.2 This will help bridge the gap between the linguistic and philosophical literature.
In the first part of the article, we offer a precise reconstruction of the mereolog- ical approach and its semantic background, in order to enable a more meaningful debate. We focus on the best-known implementation of the approach—that of Godehard Link—but most of our discussion also applies to alternative implementa- tions found in linguistics.
In the second part, we deal with the criticisms and assess their logical, linguistic, and philosophical significance. The literature contains a number of objections against the mereological approach, but it is not always clear whether (and, if so, how) they are related. We use the distinction between model theory and truth theory to disentan- gle and make more precise these objections. We contend that, upon analysis, there are four main objections, and we show how each can be addressed. Our defense is based on various considerations. For example, we emphasize that the relevant approach relies on a distinctive atomistic mereology. And we recommend that proponents of the mereological approach carefully distinguish certain object language notions from parallel notions in the metalanguage, following a broadly Tarskian strategy. We also note that counterparts of some of these objections can be raised against plural logic—
a point that can be appreciated once the distinction between model theory and truth theory has been clearly drawn. After comparing the mereological approach with plu- ral logic, we conclude that the former remains a viable and well-motivated framework for the analysis of plurals.
2 The Mereological Approach to Plurals
Mereology is the study of parthood relations. Its formal development provides a framework for comparing various theories of parts and wholes (see Simons [75] and Cotnoir and Varzi [13]). Each theory is based on a primitive mereological notion. For instance, we may start with a notion of improper parthood (≤) according to which
2For an earlier defense, see Nicolas [54].
every object is part of itself. Using this primitive, we can define other notions, such as overlap, atomicity, and the sum of several entities (see Section2.4).
The strength of a particular theory depends, of course, on the chosen axioms.
These are often expressed in the standard language of first-order logic. A well-known set of axioms yields the so-calledClassical Extensional Mereology(also known as General Extensional Mereology, see Varzi [80, sec. 4.4]). From the model-theoretic point of view, this theory describes complete Boolean algebras with the bottom element removed (this was essentially shown in Tarski [78]).3
The mereological analysis of plurals has two main components, each correspond- ing to a basic desideratum for a compositional semantics of natural language (Dowty et al. [15, pp. 44-46]). The first component is a model theory: it offers an account of the logical properties of sentences containing plurals, such as entailment, consis- tency, and equivalence (see Section2.4for examples). The second is a truth theory:
it offers an account of the truth conditions of these sentences. Details about these components are given below. As we will see, a particular theory, namely a version of Classical Extensional Mereology, is central to both components. Logical properties are captured by translating natural language sentences containing plurals into a for- mal language governed by this theory. Moreover, after assuming in the metalanguage that the axioms of the theory are satisfied, the truth conditions are given by con- straining the interpretations of natural language sentences so that plural terms denote mereological sums. Crucially, the metalanguage contains only singular expressions.
2.1 Motivations
An adequate semantics of natural language should explain the main properties of plural expressions. First, the semantics should characterize the various interpreta- tions that sentences containing plurals can receive, including their collective and distributivereadings. Here is an example using the verb phraselifted the table:
(1) Annie, Bonnie, and Connie lifted the table.
This sentence can mean that Annie, Bonnie, and Connie lifted the table together.
This is the collective reading. But the sentence can also mean that each of them lifted the table alone. This is the distributive reading. Some verb phrases only accept distributive interpretations:4
(2) Annie and Bonnie ran.
3More detailed discussions of Classical Extensional Mereology can be found in Pontow and Schubert [62], Hovda [26], and Cotnoir and Varzi [13, ch. 2].
4There may be another kind of reading, one that counts the sentence as true, for instance, when Annie and Bonnie lifted the table together, while Connie lifted it alone. This would be an intermediate reading between the collective reading and the distributive one. Accounting for readings of this kind raises a number of issues which, though interesting, will be set aside for the purposes of our discussion (see especially Gillon [20], Lasersohn [36], Schwarzschild [72], Landman [33], and Champollion [10]).
This sentence can only mean that each of Annie and Bonnie ran. We call these verb phrasesdistributive.
Second, the semantics should account for the fact that pluralsrefer cumulatively (Bunt [6, pp. 254, 262], Link [40, p. 303]). Speaking loosely, adding guests to guests gives you more guests. This property is exhibited by the following, valid inference:
(3)
The students are guests The teachers are guests
The students and the teachers are guests
As will become clear, the algebraic structure of the underlying domain enables the mereological approach to account for the readings of plural sentences and for the fact that plurals refer cumulatively.
Third, it has long been observed that plurals and mass nouns have common proper- ties.5For example, mass nouns, just like plurals, can have collective and distributive readings.
(4) The furniture is heavy.
This sentence can mean that the furniture as a whole is heavy, but it can also mean that each piece of furniture is heavy. Moreover, mass nouns, like plurals, refer cumu- latively (Quine [64, p. 83]). Again speaking loosely, adding water to water gives you more water. Furthermore, some constructions only combine with plurals and mass nouns, not with singular count nouns. For example, comparative phrases such as more catsandmore woodare grammatical, whereas more catis not. The same is true of the proportional quantifiermost:most catsandmost woodare grammatical, whereasmost catis not.6In order not to miss significant generalizations, we should seek as much as possible to explain these common features in a unified way. By assigning similar algebraic structures to the denotations of plurals and mass nouns, the mereological approach can provide a high degree of unification.
Finally, an adequate semantics of plurals should account not only for the prop- erties that plurals share with mass nouns, but also for the properties that they share with mass nounsandsingular count nouns. Indeed, these three types of expressions combine in the same way with other types of grammatical expressions, including adjectives and verbs, several determiners (e.g.the,some,any, andno), and partitive constructions:
(5) The furniture/desk/chairs is/are heavy.
(6) Annie sold the/some furniture/desk/chairs.
(7) Connie cleaned part/some/half/most of the furniture/desk/chairs.
5Overviews can be found in Link [40], Gillon [20,21], Moltmann [50], Nicolas [53,56], Pelletier and Schubert [61], Chierchia [12], Carrara and Moltmann [8], Champollion [9], and Rothstein [70].
6Some proportional quantifiers have different forms for plurals and mass nouns, e.g.many/much,few/a lit- tle. Still, they distinguish between these expressions and singular count nouns, combining with the former but not with the latter.
These data are easier to explain if we assume that the entities denoted by plurals, mass nouns, and singular count nouns belong to a single domain. The mereologi- cal approach makes precisely this assumption: the framework simply enriches the domain withspecial entitiessatisfying mereological axioms. Some versions of this approach assume separate mereologies for plurals and mass terms (e.g. Link [40,41], Landman [32], and Champollion [9]), while other versions assume a single mere- ology for both (e.g. Gillon [20], Chierchia [11], Bale and Barner [2], Sutton and Filip [76], and Landman [34]). In any case, the new entities, just like the remain- ing entities in the domain, become available for the set-theoretic operations that yield other semantic values. So the framework is easily integrated with semantics as customarily developed in linguistics.
Let us summarize the desiderata we have discussed. In general, a semantics of plurals should account for the logical properties of sentences and for their truth conditions. In particular, it should capture the distinction between collective and dis- tributive readings, the phenomenon of cumulative reference, the properties shared by plurals and mass nouns, as well as those shared by these two classes of expressions and singular count nouns. Moreover, it should be easy to integrate with the rest of semantics.
Our discussion will focus on a basic implementation of the mereological approach to plurals along the lines of Link [40], Landman [32, ch. 7] and Link [41, chs. 2, 6].
This implementation is based onindividual mereology, an atomistic mereology that uses a primitive relation calledindividual parthoodby Link. This relation must be distinguished from more familiar relations such as material parthood. For example, in the material sense, the roof of a house is part of the house, and neither of them is atomic. In contrast, individual mereology takes both the roof and the house to be atoms, ignoring their material parts and the fact that one is a material part of the other. According to the mereological approach, singular count nouns (such ashouse androof) are true of atoms in the individual sense, while plurals are true of sums of atoms in this sense. Adopting individual mereology is consistent with adopting other mereologies, including atomless ones. Multiple mereologies can live side by side.7 The mereological approach to plurals simply holds that individual mereology is the appropriate one for the analysis of plurals. As we will see, an illicit assimilation of different mereologies (e.g. individual and material) is at the heart of one of the main objections to the mereological approach (Section3.1).8
7For an example of how different mereologies may coexist, see Simons [75, chs. 1-3]. Simons adopts three parthood relations: one for ordinary objects and their parts, one for portions of matter and their parts, and one for classes and their parts. Link uses only two parthood relations, roughly corresponding to the last two used by Simons.
8It may also be useful to note that there is another notion of atomicity in the literature. A count noun is said to have atomic (or quantized) reference when the following holds with respect to a salient relation of parthood associated with the noun: if the noun is true of something that has proper parts, it is not true of any of those parts (see for instance Krifka [29], Gillon [20], Champollion [9], and Rothstein [70]). Thus, the count nouncirclehas atomic reference. It applies to a whole circle, but not to any of its proper parts.
By itself, adopting individual parthood in the analysis of plurals leaves open the question whether count nouns have atomic reference. Nicolas [53, ch. 4] argues for a negative answer to this question.
There are various reasons why we focus on a mereological approach based on individual mereology. To begin with, it has been very influential and some recent work retains its core features (e.g. Champollion [9]). Moreover, it has received a fully rigorous development, which makes easier a proper assessment of the philo- sophical criticisms. Furthermore, most of these criticisms have been stated with this framework in mind, though several of them can be reformulated so as to apply to alternative implementations of the mereological approach, including those mentioned above (Gillon [20], Chierchia [11], Bale and Barner [2], Sutton and Filip [76], and Landman [34]). Our responses to the criticisms can also be useful in the context of these alternative implementations.
We now present a basic formal framework for individual mereology. We hope that our exposition will be of interest even to readers who are acquainted with the literature. First, we offer a simplified and accessible overview of the central aspects of the framework. Second, we highlight some important methodological features. In particular, we explain how both a model theory and a truth theory can be associated with the framework. This will provide a suitable context in which the criticisms can be meaningfully reconstructed.
2.2 Semantic Framework
From the perspective of the semantic tradition stemming from the work of Richard Montague [52], semantics can be seen as involving three languages: a fragment of natural language, a formal language, and a metalanguage. Here, we are concerned with a simple fragment of English containing plurals. The formal language is used to specify the logical form of sentences of this fragment of natural language. On the mereological approach, the formal language is a first-order language together with some special vocabulary. The metalanguage is a fragment of English that includes set-theoretic and mereological vocabulary as well as the semantic notions of truth and satisfaction. We use it to formulate a model theory and a truth theory for the formal language. Crucially, the metalanguage is free of plurals on the mereological approach.9
One could try to dispense with the formal language and formulate a model theory and a truth theory directly for the fragment of natural language under study. However, this would require that we decide on a specific syntactic theory for this fragment. It is often more convenient to work with a formal language whose semantics can be developed, to some extent, independently of syntactic issues.
The basic idea behind the semantics is this. We first translate each sentence of the fragment of natural language into a formula of the formal language expressing
9We avoid using the termobject languagesince it is ambiguous in this context. The term usually refers to the language under study. So it is equally appropriate for the fragment of natural language and for the formal language—both languages are objects of semantic analysis.
its logical form. Then, in the metalanguage, we provide a model theory and a truth theory for the formal language.
fragment of natural language formal language
metalanguage
translation
model theory truth theory
The model theory characterizes a relation of logical consequence for the formal language. This is meant to illuminate logical entailments among sentences of the fragment of natural language under study. To do so, the model theory must satisfy a condition that we calllogical correctness. That is, the translation of a natural lan- guage argument should be validated by the model theory if and only if the argument itself is logically valid.10 As presented here, the model theory aims to account for logicalentailments, such as those presented in our discussion of cumulative refer- ence and distributivity (Section2.1). By relying on meaning postulates (Carnap [7]), this framework is also able to capture other kinds of entailment involving expressions with related meanings.
The truth theory specifies the circumstances in which a sentence of the formal language is true. To serve as an indirect interpretation of natural language, the formal language with its truth theory must satisfy a condition that we callalethic correctness.
That is, a sentence of natural language is true in certain circumstances if and only if its translation is true in the same circumstances.
Below we explain the distinctive feature of Link’s mereological approach, namely the use of a relation of parthood in the formal language, constrained by the model theory and the truth theory to satisfy the axioms of an atomistic version of Classical Extensional Mereology.
2.3 Translation
Following Link [40, 41] and Landman [32], we focus on a formal language that augments the standard language of first-order logic with the following items:
(i) a relation of parthood (≤);
(ii) a special class of predicates, which we callatomic;
10The right-to-left direction of the requirement assumes that the formal language is sufficiently expressive.
If not, some logically valid arguments may not be validated by the model theory. For example, proposi- tional logic seems unsuitable to capture valid arguments that involve quantifiers in an essential way. So some valid arguments may not be validated by their propositional translation.
(iii) four special operators: a pluralization operator∗, a distributive operatorD, a summation operatorσ, and a binary summation operator+.11
In this section, we give an intuitive idea of the semantics of these items. A precise characterization is provided in the following sections.
As mentioned, the translation maps sentences of natural language to formulas of the formal language. A common way of proceeding is to assign expressions of the lambda calculus to words of open classes. The translation of complex phrases can then be obtained compositionally by combining the appropriate lambda expressions.
For example, the translation ofAnnie ran, i.e.ran(a), is derived as follows. First, we mapAnnietoaandrantoλx.ran(x):
Annie → a
ran → λx.ran(x)
Given the syntax of Annie ran, we obtain its translation by composition and conversion:
Annie ran Annie ran
λx.ran(x)(a)[= ran(a)]
a λx.ran(x)
This process depends on the choice of a particular syntactic analysis of the sen- tences of the fragment of natural language under study. To avoid complications, we make only minimal assumptions and settle on formalizations that can be recovered compositionally from standard theories of syntax.
Let us provide some details about the translations we will use, while keeping in mind that the meaning of the special vocabulary will be fixed by the model theory and by the truth theory. So it is only after these theories have been presented that the choice of translations will be vindicated.
First of all, atomic predicates translate singular count nouns and singular uses of distributive verbs:
student → λx.student(x) ran → λx.ran(x)
This means that predicates likestudentandranare true only of individual atoms.12 The pluralization operator ∗ captures the semantic effect of the plural morpheme -s:
students → λx.∗student(x)
So, while student applies to any individual student, ∗student applies to any mereological sum of students.
11While the symbol+is typically used in philosophy, the symbol⊕is often used in linguistics.
12On this approach, count nouns are interpreted as atomic predicates. However, one might have to counte- nance exceptions to this rule. In particular, measure nouns such asliterandkiloseem to require a special semantic treatment—see Champollion [9, ch. 7], Rothstein [70, chs. 9-10], and references therein.
The summation operatorσcaptures the meaning of the definite article:
the student → σ x.student(x) the students → σ x.∗student(x)
Essentially, the operator yields the mereological sum of the entities satisfying the formula bound byσ. Note that if there is only one student,σ x.student(x)denotes this student.
The binary summation operator+corresponds to nominal conjunction:
Annie and Bonnie → a+b
In this example, it yields the mereological sum of the denotations ofaandb.
Finally, the distributive operatorD is inserted when a verb phrase is understood distributively:
Annie and Bonnie won→won(a+b) (collective reading) Annie and Bonnie won→D won(a+b) (distributive reading)
Intuitively,won(a+b)means that the denotation ofa+bwon as a team, whereas
Dwon(a+b)means that the denotations ofaandbindividually won.13
In the next section, we describe the model theory associated with the formal language used by the translation.
2.4 Model Theory
The model theory characterizes the relation of logical consequence for sentences of the formal language and hence, indirectly, for those of natural language. There are a number of reasons why a fully worked-out model theory is important. To begin with, such a theory is essential for determining whether the semantics captures the logical properties of plurals. Some of the phenomena described in Section2.1, such as cumulative reference and distributivity, manifest themselves through logical relations. To capture these relations a model theory is required. Furthermore, some objections against the mereological approach involve logical considerations that can only be assessed once a model theory has been properly formulated (see especially the argument from incorrect existential consequence in Section3.3).
In model theory, logical consequence is defined as truth preservation in every model of the language. Truth in a model is defined inductively using the appropri- ate features of the model. In this setting, a model is a structureM=
D, A, ,· . The first component,D, A, , is a complete atomic Boolean algebra with the bot- tom element removed:Dis the domain;A, a subset ofD, is the set of atoms; and
13There is disagreement about exactly which linguistic phenomena require the use of such a distributive operator. For discussion, see e.g. Champollion [10, pp. 289-308].
is the operation of join.14The canonical partial order associated with this structure is defined as follows:
xy =def xy=y
This relation provides the interpretation of individual parthood.
The second component of the structure,·, is an interpretation function assigning denotations to various expressions in the language. It satisfies the usual constraints.15 For example, the denotation of a term is an element of the domain, whereas the denotation of a predicate or an open formula is a set. In addition, the interpretation function satisfies constraints that ensure the intended semantics of parthood, atomic predicates, and the four special operators. An intuitive idea of this semantics was given in the previous section. We now offer a more precise characterization.16
The special relation≤is always interpreted as, the canonical partial order of the algebra. This ensures that≤satisfies the axioms of Classical Extensional Mereology in every model of the language. In other words,≤stands for a partial order satis- fying the axiom of Strong Supplementation and the axiom scheme of Unrestricted Composition:
(Strong Supplementation) ∀x∀y(¬x ≤y→ ∃z(z≤x∧ ¬z◦y))
(Unrestricted Composition) ∃x ϕ(x)→ ∃z∀w(w◦z↔ ∃x(ϕ(x)∧w◦x)) where the symbol◦stands for overlap, namelyx◦y =def ∃z(z≤x ∧ z≤y).17
Given its interpretation as, the relation of parthood satisfies an additional mere- ological condition, namely Atomicity. Let us say thatxis an atom if it has no part other than itself:
atom(x)=def∀y(y≤x → y=x) Then Atomicity states that every object has an atom among its parts:
∀x∃y(atom(y) ∧ y≤x)
We require thatatomic predicatesdenote either a set of atoms in the algebra or the empty set.
Let us now specify the constraints on the interpretation of the four operators intro- duced and intuitively explained in the previous section. The pluralization operator∗ applies to any atomic predicateN. If the denotation ofNis not empty, the denotation of∗Nis the closure under join () of the denotation ofN. Otherwise the denotation of
∗N is empty. The fact that plurals refer cumulatively follows from this interpretation of the operator∗. We will see an example shortly.
14Given Stone’s representation theorem, any complete atomic Boolean algebra is isomorphic to a powerset algebra. So we can think of a complete atomic Boolean algebrawith the bottom element removedas a substructure of a powerset algebra. More precisely, we start with a setUand obtain the algebraic structure P(U )− ∅, A,∪ . The domain of the algebra isP(U )− ∅, the set of atomsAis{{x} :x ∈U}, and the join is the operation of set-theoretic union. Since the empty set has been removed, this structure is not closed under intersection and thus the operation of meet is only partial.
15See, e.g., Enderton [16, sec. 2.2].
16Since we focus on the interpretation of plurals, we may ignore complications arising from the analysis of linguistic phenomena such as tense and mood. As a result, there is no need to introduce time indices or possible worlds in the model theory.
17See Varzi [80, sec. 4] for more context and discussion.
The distributive operatorD applies to predicates of any kind and its interpreta- tion is constrained as follows. Something is in the denotation ofDP if and only if every atomic part of it is in the denotation ofP.18That is, every model satisfies this formula:
∀xD
P (x) ↔ ∀y(atom(y)∧y≤x → P (y))
Finally, we specify the interpretation of the summation operators. Letϕ(x)be any formula wherexoccurs free. Applying the operatorσyields the termσ x.ϕ(x)where xis no longer free. If the denotation ofϕ(x)is not empty and contains its own join, then the interpretation ofσ x.ϕ(x)is this join. Otherwiseσ x.ϕ(x)fails to denote.19 As for the binary operator+, it is interpreted as. This means thata+bdenotes the join of the denotations ofaandb.
The model theory outlined in this section captures a variety of inferences. For example, it captures the logical relations characteristic of distributivity:
(8) Annie and Bonnie run Drun(a+b)
Annie runs and Bonnie runs → run(a) ∧ run(b)
The validity of this inference can be verified as follows. Suppose that the premise is true in a model, that is, the denotation ofa+bis in the denotation ofDrun. The interpretation of the distributivity operator ensures that the atoms denoted byaand bare both in the denotation ofrun. This reasoning makes the common assumption that a constant translating a singular proper name denotes an atom.
Similar algebraic reasoning explains why plural expressions like guests refer cumulatively:
(9) The students are guests ∗guest(σ x.∗student(x)) The teachers are guests → ∗guest(σ x.∗teacher(x)) The students and the teachers are guests ∗guest(σ x.∗student(x)+
σ x.∗teacher(x)) Given the interpretation of the pluralization operator∗, the denotation of∗guest is closed under join. So if it contains the denotation ofσ x.∗student(x)and that of σ x.∗teacher(x), it also contains the denotation ofσ x.∗student(x)+σ x.∗teacher(x).
In this system, the key relation ofbeing one ofamounts to the relation of being an atomic part. So the sentenceAnnie is one of the studentsis translated as:
atom(a) ∧ a≤σ x.∗student(x)
18As mentioned, this operator is used for the distributive readings of verb phrases. To account for the inter- mediate readings mentioned in footnote4, one can introduce a more general version of the operator (for discussion, see Gillon [20, pp. 617-620], Schwarzschild [72, pp. 68-71], and Champollion [10, sec. 4.4]).
Allowing for this generalization is one reason to distinguish the pluralization operator∗and the distribu- tive operatorD. The former applies to the translation of noun phrases, the latter to the translation of verb phrases.
19Note that, on this interpretation, the summation operationσ does not correspond exactly to the join operation, since it adds the extra condition that the join be in the denotation ofϕ(x). This allowsσto capture the meaning of the definite article as it combines with both singular and plural count nouns (see Landman [32, pp. 305-306]).
It is easy to see how this translation captures intuitive inferences such as the following:
(10) Annie is one of the students atom(a) ∧ a≤σ x.∗student(x)
Annie is a student → student(a)
Suppose that the premise is true in a model. The interpretation ofσ and the fact thatstudentis an atomic predicate guarantee that the denotation ofamust be in the denotation∗student. Since the denotation ofais an atom andstudentis atomic, the algebraic closure effected by the pluralization operator∗entails that the denotation ofais also in the denotation ofstudent.
We have provided a basic characterization of the model theory, and we have ver- ified that it captures fundamental properties of plurals, such as distributivity and cumulative reference. We will now turn to the formulation of the truth theory.
2.5 Truth Theory
Proponents of the mereological approach to plurals such as Link [40], Landman [32], and Gillon [20] provide only a model theory. However, some philosophical objections do not make much sense in the context of a model theory and have bite only with respect to a truth theory. So we explain how one may develop a truth theory that complements the model theory presented above. We introduce a primitive relation of individual parthood in the metalanguage and assume that it satisfies the mereological axioms just mentioned. We then use individual parthood to interpret the relation≤of the formal language. This ensures that≤has the appropriate mereological behavior.
The basic principles of the truth theory mirror those of the model theory.20 For instance, one requires that an atomic predicateN be satisfied only by mereological atoms, and that∗N apply to any sum of those atoms. Furthermore, suppose that ϕ(x)is satisfied. Thenσ x.ϕ(x) denotes the sum of everything that satisfies ϕ(x) provided that this sum also satisfiesϕ(x). Otherwiseσ x.ϕ(x)fails to denote. Given the interpretation of+, a+bdenotes the sum of the denotations ofa andb. As before, distributive readings are captured by a constraint on the interpretation of the operatorD. For any predicateP, something satisfiesDP if and only if every atomic part of it satisfiesP.
To see how truth conditions are specified, consider the collective readings of the next two sentences (with people winning together as teams), followed by their translation:
(11) Annie and Bonnie won.
(12) The students won.
(13) won(a+b)
20The literature contains two ways of framing the truth theory. One regards predicates as non-denoting expressions (see Larson and Segal [35, ch. 4]). The resulting truth theory is developed independently of the model theory. The other option, which is the most popular in linguistic semantics, assigns denotations to predicates. The truth theory can then be viewed as the result of applying the model theory to a model corresponding to the actual world. Our presentation is meant to be consistent with both options.
(14) won(σ x.∗student(x))
The truth conditions of (13) and (14) are given by (15) and (16), respectively:
(15) won(a + b) is true if and only if the denotation of a +b satisfies the predicatewon.
(16) won(σ x.∗student(x))is true if and only if the denotation ofσ x.∗student(x) is not empty and satisfies the predicatewon.
Given the principles stated above, the background mereology ensures that these truth conditions are equivalent to the following:
(17) won(a+b)is true if and only if the sum of the denotations ofaandbsatisfies the predicatewon.
(18) won(σ x.∗student(x))is true if and only if something satisfies the predicate
∗studentand the sum of everything that satisfies this predicate satisfies the predicatewon.
This concludes our exposition of the mereological approach. In the next section, we describe its main competitor within philosophical logic, namely plural logic.
Being able to compare the two frameworks will be important when we turn to the objections to the mereological approach.
2.6 Plural Logic
In its most common form, plural logic represents plural predication in natural lan- guage using a type distinction that reflects the grammatical distinction between singular and plural. The formal language is obtained by expanding first-order logic with new types of variables, quantifiers binding them, and predicates. So there is a sharp distinction between singular and plural variables, quantifiers, and predicates.
Consider the following sentences in their collective readings (note that, in this paper, we adopt the philosophical use of the wordthing, which does not presuppose inanimacy):
(11) Annie and Bonnie won.
(19) Some things won.
In plural logic, they can be translated as follows:
(20) WON(a&b) (21) ∃xx WON(xx)
wherea&bis a plural term conjoining two singular terms,WONis a plural predicate, andxxis a plural variable bound by a plural existential quantifier.
It is technically possible to implement the model theory for this formal language in set theory by assigning plural variables a domain with precisely the same algebraic structure as the one used for the mereological approach (see, e.g. McKay [47, ch. 5]).
However, it is far more popular to formulate the model theory by adopting primitive plural resources in the metalanguage and using them to interpret the plural variables
of the formal language (e.g. Rayo [66], McKay [47, ch. 6], Yi [86], and Oliver and Smiley [58, ch. 13]).
The main advantage of this kind of model theory is that it gets closer to the intended interpretation of the fragment of natural language under study than the set-theoretic model theory. A plural term is not taken to stand for a set. Rather, it stands for some things. Similarly, a domain of quantification can be described plu- rally as some things, those over which the quantifiers range. This appears to yield interpretations where the range of the quantifiers is not set-sized but unrestricted.
The price to be paid for this model theory is that the metalanguage has to go beyond plurals and include quantifiers of a new type (see Rayo and Uzquiano [67], Rayo [66], Yi [86], and Florio [17]), an observation that will be relevant in later sections (3.2 and 4). These new quantifiers can be of two kinds, depending on whether they bind variables in predicate position or in nominal position. Quantifiers of the former kind are calledhigher-order; quantifiers of the latter kind are often calledsuperplural.
Let us briefly explain how this kind of model theory works. Suppose we adopt higher-order quantifiers. Then we can quantify into the predicate position of plural predicates:
(22) ∃X X(a&b)
We may gloss this formula in terms of properties: there is a property jointly possessed by the denotations ofaandb. Using higher-order quantification, one can provide an interpretation of the collective reading of a plural predicate such as WON. The interpretation states that, in any model, there is a propertyX for which the plural predicateWONstands.
Superplural quantifiers can play an analogous role. Intuitively, they enable quan- tification over “pluralities of pluralities”:
(23) ∃xxx (a&bare amongxxx ∧ c&dare amongxxx)
This formula asserts that the denotation ofa&band the denotation ofc&dare among some plurality of pluralities. The variablexxxis superplural, bound by a superplural quantifier∃xxx. Quantifiers of this kind can be used to provide an interpretation of the collective reading of plural predicates. The interpretation states that, in any model in which the predicateWON is satisfied, there is a superpluralityxxx for which it stands.
The ascent to resources of a new type is not needed for a truth theory. One can simply add to the metalanguage a new plural predicate for satisfaction. As a result, one may give the following truth conditions for the examples above:
(24) WON(a&b)is true if and only if the denotations ofaandbcollectively satisfy the plural predicateWON.
(25) ∃xx WON(xx)is true if and only if some things collectively satisfy the plural predicateWON.
This completes our exposition of the core aspects of the mereological analysis of plurals and of plural logic. We are now ready to engage in an informed debate about the objections leveled against the analysis.
3 Objections
The literature contains a wealth of objections against the mereological approach, and it is not always clear whether (and, if so, how) they are related. In the rest of the article, we pursue two goals. The first is to offer a clear and useful picture of this landscape, which will help readers, both in philosophy and linguistics, orient themselves. The second is to assess the logical, linguistic, and philosophical import of the most significant objections. The methodological clarity provided by a precise formulation of the mereological approach is crucial in the pursuit of these goals.
We contend that the most significant objections can be divided into four kinds.
The characteristic feature of objections of the first kind is that the relation of part- hood used to analyze plurals is identified with some other mereological relation, e.g.
material parthood. We call themflatteningobjections. The second kind of objection involves sentences featuring mereological notions. We refer to them asreflexivity objections. The third kind of objection concernsontological commitment. Finally, there is an objection that questions theintelligibility of plural predication on the mereological account.
As presented above, the mereological approach has two main components: a model theory and a truth theory. By distinguishing between the two, a sharper for- mulation of the objections can be provided. This also puts the proponent of the mereological approach in a better position to respond. Once the distinction is drawn, the first two kinds of objection can be viewed as targeting both theories. (For ease of exposition, we present the version concerning the truth theory, which is easier to appreciate.) However, it also becomes clear that objections of the third and fourth kind have force only with respect to the truth theory. Furthermore, the distinction between model theory and truth theory reveals that counterparts of some of these objections can be raised against plural logic.
3.1 Flattening
Flattening objections exhibit the following pattern. Plurals are analyzed using a mere- ological relation other than individual parthood, and this is shown to have implausible consequences. Here is an example. Agust´ın Rayo writes:
Suppose that there are a few scattered piles of sand on the table. Then it is true of the piles of sand, but false of the grains of sand which make up the piles, that they are scattered. But, if we take mereological sums to be our surrogates, this fact cannot be captured [...], since the mereological sum of the piles is pre- cisely the same object as the mereological sum of the grains of sand. (Rayo [65, pp. 444-445])
The objection begins with sentences (26) and (27), and their translations:
(26) The piles of sand are scattered.
(27) The grains of sand are not scattered.
(28) scattered(σ x.∗pile-of-sand(x)) (29) ¬scattered(σ x.∗grain-of-sand(x))
In Section2.2, we noted that a successful translation should satisfy what we called alethic correctness. Thus, in the envisioned scenario, the translation should satisfy these two conditions:
(i) (26) is true if and only if (28) is true;
(ii) (27) is true if and only if (29) is true.
According to the truth theory outlined in Section2.5, the termσ x.∗pile-of-sand(x) denotes the sum of everything satisfying the predicatepile-of-sand(x). Likewise, the termσ x.∗grain-of-sand(x)denotes the sum of everything satisfying the predicate grain-of-sand(x). The argument makes the key assumption that these sums are iden- tical. As a result, sentences (28) and (29) are taken to have opposite truth values: the first sentence says that this sum satisfies the predicatescattered(x), whereas the sec- ond denies it. So it follows from conditions (i) and (ii) that (26) and (27) also have opposite truth values. But this contradicts what is stipulated in the example, namely that the piles of sand are scattered while the grains of sand are not.
In response to the objection, we should deny the key assumption that the sum denoted by σ x.∗pile-of-sand(x) is the same as the sum denoted by σ x.∗grain-of-sand(x). The assumption is plausible on a material reading of the notion of sum. On this reading, the assumption amounts to the claim that the stuff that makes up the piles of sand is the very same stuff that makes up the grains of sand. However, this understanding of the notion of sum is not the one operative in the semantics. The semantics is based on individual mereology, relative to which each pile of sand is an atom and each grain of sand is an atom. Two individual sums of atoms are identical just in case they have the same atoms. It follows that the indi- vidual sum of everything satisfying the predicatepile-of-sand(x)is distinct from the individual sum of everything satisfying the predicategrain-of-sand(x). Of course, the individual mereology operative in the semantics is consistent with the existence of other mereological relations. As we remarked earlier, an individual sum can have many material parts that do not correspond to its decomposition into individual atoms.21
An alternative response may be available if the semantics of expressions likepile of sanddiffers from that of simple count nouns (see, for example, Schwarzschild [73], Pearson [60], Champollion [9], and Rothstein [70]). However, a response of this kind would not block other flattening objections, for example one put forward by Tom McKay [47, p. 42]. This objection involves regions of space.
21This version of the objection concerns the truth theory. One can formulate a version that targets the model theory by focusing on theconsistencyof sentences (26) and (27). Assuming that the model theory interprets the subjects in the same way, the translations of these sentences become inconsistent.
There is a sense of parthood that applies to regions of space: a region of space can be part of another, and two regions can overlap, i.e. have a part in common. Now consider two regions,a andb, that overlap. Leta−r be the largest part ofathat does not overlapr. Likewise, letb−rbe the largest part ofbthat does not overlapr.
Then these two collective predications are true:
(30) Regionsaandboverlap.
(31) Regionsa−r,b−r, andrdo not overlap.
They are translated as follows:
(32) overlap(a+b)
(33) ¬overlap((a−r)+(b−r)+r)
Relative to the mereology of spatial regions, the sum of the regionsaandbis identical to the sum of the regionsa−r, b−r, and r. So the term a +b and the term (a−r)+(b−r)+rdenote the same region, namely this sum. Then, (32) and (33) have opposite truth values, unlike the sentences they translate.
It is easy to see that our response to the preceding example applies here too. The sense in which the sum of the regionsaandbis identical to the sum of the regions a−r,b−r, andrinvolves the mereology of spatial regions. But, again, this is not the notion of sum operative in the semantics. With respect to individual parthood, each region is a distinct atom. Therefore, the individual sum ofa−r,b−r, andr(three distinct atoms) cannot be identical to the individual sum ofaandb(two other atoms).
Objections of this kind are an important reminder that a successful analysis of plurals must respect the distinction between individual parthood and other notions of parthood. Flattening otherwise distinct mereological relations has implausible consequences for the semantics.
In one of their early articles on plurals, Alex Oliver and Timothy Smiley also put forward a flattening objection:
wholes [...] can be decomposed into parts in many ways. This is why mereo- logical sums or fusions are ineligible [to analyze English plurals]. For example,
‘Whitehead and Russell’ and ‘the molecules of Whitehead and Russell’ rep- resent different decompositions of the same sum, but giving them that sum as their common reference forces the conclusion that the molecules of Whitehead and Russell were logicians. (Oliver and Smiley [57, p. 293])
They acknowledge the response offered above but contend that the appeal to indi- vidual parthood, which they call “the artificial mereology of ‘lattice-theoretical’
semantics”, is still problematic. They claim that the mereological approach cannot provide a satisfactory semantics for the following sentence:
(34) The individual sums are more numerous than the individual atoms.
However, there is a crucial difference between this case and the previous examples:
(34) explicitly involves mereological notions in the individual sense, while the pre- vious examples do not. So (34) presupposes that the fragment of natural language under study contains the very mereological notions that are used in the semantics.
This presupposition is the defining feature of the second kind of objection, to which we now turn.
3.2 Reflexivity
Assume for the moment that the fragment of natural language under study contains the very mereological notions that one wishes to use in the semantics. In our case, this is individual mereology. Then, as a number of authors have suggested, some sentences seem to receive an incorrect analysis. We call this phenomenonreflexiv- ityto emphasize that these cases arise when the mereological vocabulary present in the metalanguage is reflected, and hence incorporated, into the fragment of natural language under study.
As indicated in Section2.4, it is common to assume that a constant translating a singular proper name denotes an atom with respect to individual parthood. For the sake of argument, suppose that this assumption does not apply to the proper names and, hence, thatsdenotes a non-atomic sum. Now consider this simple sentence:
(35) sis a sum.
The sentence is true. According to the semantics, its translation sum(s) is true if and only if the denotation ofssatisfies the predicatesum. But, recall, the semantics requires that any singular count noun be translated as a predicate satisfied only by atoms. Because of this constraint,sumcannot be true of a non-atomic sum. So the semantics cannot account for the truth of (35), violating alethic correctness. (This objection can be found in Moltmann [50, pp. 18-19].)
Next, consider an example inspired by Barry Schein [71, pp. 33-37], which raises a problem even if we suppose that the translation of certain singular count nouns (such asnon-atom) can be true of non-atomic sums:
(36) The atoms are the non-atoms.
This sentence is intuitively false. Its mereological translation is assumed to be (37):
(37) σ x.∗atom(x)=σ x.∗non-atom(x)
Here the predicatenon-atomis taken to be satisfied by any non-atomic sum. Given the axioms of individual mereology, both terms flanking the identity sign in (37) denote the sum of everything satisfying the predicateatom. So (37) is true but (36) is not, which violates alethic correctness.22The example given by Oliver and Smiley, i.e. sentence (34) above, can also be seen as problematic for similar reasons.
How can reflexivity objections be avoided? We think that a simple response is available to the defender of the mereological approach. This is to insist that the mere- ological notions used in the semantics benew, namely that they not be part of the language under study. Is this simple response too good to be true? We think not. Let us counter three objections that might be raised against this response.
22The objection can also be formulated so as to target the model theory. This is done by focusing on the intuitiveinconsistencyof (36).
First, one might object that the new mereological resources are unintelligible and thus unsuitable for semantic theorizing. Can we make sense of them? Consider the truth and satisfaction predicates used in the metalanguage. We are working within a Tarskian framework in which these are assumed to be new with respect to the lan- guage under study (Tarski [77]). We make sense of these new predicates by laying out the axioms that govern them. And we make sense of the new mereological resources in exactly the same way. Proponents of plural logic face a similar issue. They, too, must introduce new resources. As noted in Section2.6, their metalanguage involves new truth and satisfaction predicates as well as quantifiers of a new type.23
Second, one might object that the response we advocated makes plural talk of atoms and sums semantically unanalyzable. Suppose that these notions figure as count nouns in the singular metalanguage. Nothing, it appears, prevents us from plu- ralizing them and speaking ofatomsandsums. One might expect the mereological approach to be able to give an analysis of these expressions. However, our response seems to make this impossible. For it requires the mereological notions used in the analysis of plurals to be new with respect to the language under study. So, if this language already contains these notions, the analysis offered above becomes unavail- able. The obvious way out is to exploit the Tarskian approach embraced here, and to introduce new resources in the (meta-)metalanguage. An adequate analysis of plural talk ofatomsandsumscan be given by relying on new mereological notions, such assuper-parthood,super-sumandsuper-atom. This step must then be iterated if one wishes to analyze plural talk involving these new notions. The iteration continues, and the effect is an ontological hierarchy of mereological levels, where the sums at one level are the atoms at the next level.
Third, one might object that the new expressive resources are not economical. The truth theory of the mereological approach introduces new truth and satisfaction pred- icatesas well asnew mereological notions. In contrast, the truth theory of plural logic relies only on new truth and satisfaction predicates. So why should one introduce more notions than strictly necessary? One reply in defense of our position is that the introduction of new mereological resources leads to an empirically adequate account of a variety of natural language phenomena (Section2.1).
Moreover, it is important to be careful when assessing the expressive economy of competing frameworks: one should consider not only the truth theory, but also the model theory. Take the case of plural logic. Recall that the most popular model theory for this formal language goes beyond plurals and introduces quantifiers of a new type (Section2.6). So, if one wants to analyze this new language, one needs to introduce yet another type of quantifier. The introduction of new resources con- tinues, resulting in a type-theoretic hierarchy analogous to the ontological hierarchy postulated by proponents of the mereological approach (see, e.g., Rayo [66], Lin- nebo and Rayo [45], and Florio and Linnebo [18, ch. 11]). We come back to this
23This Tarskian framework, which relies on classical logic, is typically adopted by proponents of both the mereological approach and plural logic. Rejecting this framework opens up a number of possibilities that have been thoroughly studied in the literature on classical and non-classical theories of truth (see, e.g., Halbach [23] and Beall et al. [3]). It would be interesting to explore these possibilities in the context of the present debate. However, that would go much beyond the scope of this article.
issue in Section4, where we stress the benefits of the ontological hierarchy over the type-theoretic one.
For the moment, let us remark that the introduction of new levels of quantifiers raises difficult issues. To begin with, their intelligibility and legitimacy has been questioned. Famously, Quine rejected the use of higher-order quantifiers (Quine [63, pp. 66-68]). More recently, a number of authors have expressed doubts about super- plurals (e.g. McKay [47, pp. 46–53] and Ben-Yami [4]). Moreover, a longstanding criticism is that adopting a hierarchy of such quantifiers gives rise to expressive lim- itations (see Linnebo [43] and Kr¨amer [28] for a recent discussion). In this respect, the mereological approach is more economical, since its metalanguage contains only one type of quantifier.
Let us take stock of where we are so far. We have advocated a simple response to reflexivity objections, namely to usenewmereological notions in the semantics, made intelligible through axioms. As we have seen, this response does not prevent one from giving a semantic analysis of talk ofatoms andsums, if one wishes to do so. The new expressive resources are theoretically motivated, and they are not obviously less economical than those of plural logic. So we find no compelling way to substantiate the worry that the simple response is too good to be true.
Once it is recognized that adding new mereological vocabulary is legitimate, we have the resources to deal with another important argument against the mereological approach and, more generally, against singularist analyses of plurals. The argument is inspired by Russell’s paradox (see Boolos [5, pp. 440-441], Lewis [39, p. 65], Schein [71, ch. 2], Higginbotham [24, pp. 16-17], Oliver and Smiley [57, pp. 303- 304], and Rayo [65, pp. 439-440]). It is easier to see the problem when the singularist uses sets to translate plurals. Considering this sentence:
(38) There are some sets such that any set is one of them if and only if it is not a member of itself.
Its translation is assumed to be the following:
(39) ∃x
set(x)∧∃y(y ∈x)∧∀y(y ∈x→set(y))∧∀y(set(y)→(y∈x↔y∈y)) The problem is that (39) is inconsistent: the sentence implies that there is a set that is a member of itself if and only if it is not. Yet, (38) seems to be true.
As is clear, the argument under discussion is just another reflexivity objection.
The problematic sentence involves the very same notions that the semantics uses to analyze plurals, namely set and membership. Following the response advocated above, the singularist could introduce new set-theoretic notions (e.g.super-setand super-membership) to give an analysis of talk ofsets andmembers in the plural.
This strategy has been recently developed and defended in Meadows [49] (see also Williamson [82]). The move would enable the singularist to analyze (3.2) in this way:
∃x
super-set(x)∧ ∃y(ysuper-member-ofx)
(40) ∧∀y(y super-member-ofx→set(y))
∧∀y(set(y)→(ysuper-member-ofx↔y∈y))
This formula is not only consistent but also true if one adopts the appropriate axioms for super-sets.
The upshot is that, whether formulated in the context of set theory or mereology, arguments inspired by Russell’s paradox are just instances of reflexivity objections.
So they are blocked by the use of new semantic primitives in the analysis of plurals.
The need for such primitives is a fact of life in semantics.24Plural logic is not immune from it.
3.3 Ontological Commitment
So far, we have argued that neither flattening objections nor reflexivity objections pose a serious problem for the mereological approach. The third kind of objection we want to address has to do with ontological commitment (e.g. McKay [47, pp. 28-29]).
Let us start with one of our earlier examples of plural predication:
(41) Annie and Bonnie won.
The translation of its collective reading is:
(42) won(a+b)
And its truth conditions are:
(43) won(a+b)is true if and only if the sum of the denotations ofaandbsatisfies the predicatewon.
If (42) is true, we can assert in the metalanguage that there exists a mereological sum.
Together with alethic correctness, this implies (44):
(44) IfAnnie and Bonnie wonis true, then there is a mereological sum.
Suppose that we have a clear-enough notion of ontological commitment for natural languages sentences, and suppose that (44) expresses an ontological commitment of (41). Then one can formulate an objection to the mereological approach relying on two premises. First, the truth of (41) makes in fact no commitment to the existence of sums (or any other kind of collection). Second, an adequate semantics should always respect the commitments of the sentences it analyzes.25
Concerning the first premise, it is important to note that intuitions about these commitments tend to vary (see, e.g., Boolos [5], Resnik [68], and Landman [31]; for
24As observed in footnote 23, this conclusion depends on the assumption of a broadly Tarskian framework, which is typically made by both parties to the debate.
25One way to block the objection would be to claim that mereology is ontologically innocent and hence reject the presupposition that (44) expresses a genuine ontological commitment of (41). A number of authors, including David Armstrong, David Lewis, and more recently Achille Varzi and Rohan French, have defended the view that mereology is indeed ontologically innocent (see Armstrong [1, ch. 2, sec. 1.2], Lewis [39, ch. 3, sec. 6], Varzi [79], and French [19]). If one takes this view, a commitment to sums is no commitment at all. The relation of parthood used in the semantics of plurals is different from that discussed by these authors. However, similar arguments may be available for individual parthood (see Link [41, pp. 315-317]). Here we prefer not to take a stand on whether this kind of response is successful.
an overview, see Linnebo [44, sec. 1.5]). Some have argued that plural quantification commits us to sets or set-like collections. In a similar vein, proponents of the mere- ological approach can simply insist that their analysis reveals the true commitments of plural sentences. So we agree with Øystein Linnebo that objections based on the notion of commitment should not be given significant weight (Linnebo [42]).
Moreover, an adequate semantics need not always respect the ontological com- mitments of the sentences it analyzes. This is because failing to respect these commitments may be offset by the advantages of the semantics. The classic example is Donald Davidson’s analysis of predication (Davidson [14]). The analysis intro- duces ontological commitments to events, but it would be hard to maintain that these commitments can be recognized pre-theoretically. Yet, it is precisely the appeal to events that makes possible a simple and successful account of the semantics of adverbs. Something similar is true of the mereological approach: it provides a pow- erful way to account for important properties shared by plurals, mass nouns and singular count nouns (Section2.1).
The problem of ontological commitment concerns the truth theory. It should not be confused with another kind of objection found in the literature (McKay [47, p. 23]; see also Oliver and Smiley [57, pp. 295-296], Yi [84, pp. 165-166], and Yi [85, pp. 468-469]). This kind of objection concerns logical consequence, and hence the model theory. Singular and plural expressions can make different logical contributions to inferences. The alleged problem is that the mereological approach misrepresents those contributions. Let us consider these sentences:
(45) Annie and Bonnie won.
(46) Something won.
Intuitively, a collective reading of (45) does not entail (46): Annie and Bonnie won together, so there need not be single entity that won. The objection is that the mereological approach will validate the incorrect inference from (45) to (46).
This objection is unfounded. The bare quantifiersomethingin (46) is used in the singular. In Link’s implementation of the mereological approach, singularity corre- sponds to atomicity. So the mereological translation of (45) and (46) is given by the following formulas:
(47) won(a+b)
(48) ∃x(atom(x) ∧ won(x))
It is easy to verify that the step from (47) to (48) is invalid according to the semantics presented in Section2.4. Take a model in which the denotation ofa+b, a non-atomic sum, is the only entity in the denotation ofwon. In this model, (47) is true but (48) is false.
Moreover, if we assume that the count nounthingapplies to all and only atoms in the domain, the semantics can be shown to capture the validity of the inference from (45) to (49):
(49) Some things won.
